
We can replace it by the requirement that the solution is locally bounded away from 1, that is, for every compact subset \(K\subset \Omega _T\)  there exists a \(\mu \)  such that \(u\le 1-\mu \)  in \(K\) .
In this case, \(C\)  and \(\alpha \)  in thm:holder do depend on \(K\) .

The assumption is not restrictive and builds upon our earlier work [1]}.
There we showed that \(u\le 1-\mu \)  provided that the initial data \(u_0\)  satisfies \(u_0\le 1-\theta \) , where \(\mu \)  depends on \(\theta \) .
This argument relies on the blow-up behaviour of \(\phi \)  encoded in (REF ) and on the domain being bounded and having a Lipschitz boundary.
Now, the interior Hölder continuity of \(u\)  only depends on the initial data \(u_0\)  through \(\mu \) .

The assumption implies that the singularity of \(\phi \)  in (REF ) is not attained.
Indeed, we assume that \(u\le 1-\mu \)  so the behaviour of \(\phi \)  in \((1-\mu ,1)\)  does not matter, and therefore we could remove (REF ) provided that we impose that \(\phi ^{\prime }\ge \lambda >0\)  on \([\varepsilon ,1-\mu ]\)  for some \(\lambda >0\)  and \(\phi (0)=0\) .
Consequently, our result also holds for degenerate equations without a singularity such as the Porous-Fisher equation, see [2]}.
However, we aim to prove Hölder continuity for the solutions obtained in our earlier work on well-posedness, see [1]}, where (REF ) is a key ingredient to obtain the bound \(u\le 1-\mu \) .
Moreover, (REF ) plays a fundamental role to obtain the uniform bound \(u<1\)  and it is necessary to be able to formulate \((\ref {eq:SD-PME})\)  as a Stefan problem of the form (REF ), since otherwise \(\phi \)  might not be invertible.
Therefore, we keep the hypothesis (REF ).

